Function analysis
Determine domain, range, and key behaviors of a function from its equation or graph.
Objective
Determine domain, range, and key behaviors of a function from its equation or graph.
What it means
Analyzing a function means finding its domain (allowed inputs), range (possible outputs), intercepts, and where it increases or decreases. Domain restrictions come from denominators (cannot be 0) and even roots (radicand must be >= 0).
Key terms
- Domain
- The set of all valid x-values
- Range
- The set of all possible y-values
- Increasing interval
- x-interval where y grows as x grows
- x-intercept
- Point where the graph crosses the x-axis (y = 0)
Rules to remember
- Denominators cannot equal 0 — exclude those x-values from the domain.
- Expressions under an even root must be >= 0.
- Polynomials have domain of all real numbers.
Step by step
- 1Scan for restrictionsLook for denominators and even roots.
- 2Solve the restrictionSet denominator = 0 or radicand < 0 and solve to exclude values.
- 3State the domainWrite all real numbers except the excluded ones, or an interval.
Common mistakes
- Allowing the radicand to equal a value making a denominator 0.
- Forgetting to exclude the single point where a denominator is 0.
Tips
- Always check denominators and even roots first.
- Write domain restrictions as inequalities or ≠ statements.